The structural flexibility enabled by current holey fibre’s fabrication techniques allows the possibility of scaling the pitch Λ to dimensions ranging from the sub-wavelength to several times the wavelength, and to achieve air-filling fractions from a few percent to more than 90%. This enables fibres with very different modal extensions, and therefore nonlinear properties, to be realised. The common way to quantify the spatial extension of the guided mode is to introduce an effective area, which is a fundamental quantity in the context of nonlinear phenomena. In the derivation of the nonlinear Schr¨odinger equation for the electric field E(r), a nonlinear coefficient γ is derived. This quanti- fies the strength of the nonlinear effects and depends on both material and waveguide
properties [6]:
γ = 2π
λ n2
Aeff. (1.6)
Heren2 is the material-related nonlinear coefficient and Aeff is the effective area, which depends on the geometric parameters and is defined as
Aeff = µZ Z ∞ −∞ |F(x, y)|2dxdy ¶2 Z Z ∞ −∞ |F(x, y)|4dxdy (1.7)
withF(x, y) the modal field distribution andx and y the transverse coordinates. Conventional single mode, step-index fibres for telecoms applications present Aeff '
80 µm2, which corresponds to γ' 1 W−1km−1. Smaller values of A
eff are useful in
nonlinearity-based devices, where a strong light confinement enhances the nonlinear interactions, while a larger mode area is attractive in high power applications where it helps minimise the nonlinear effects and increase the maximum power level that can be tolerated without incurring damage. By changing the dopant profile and concentration it is possible to realise conventional single mode fibres with Aeff ranging from ∼ 10
µm2 [80] to ∼400µm2 [81].
The advent of HF technology has allowed improvements in both areas of highly nonlin- ear and large mode area fibres. Small core, large air-filling fraction HFs have permitted Aeff of a few µm2 to be realised, almost matching the minimum theoretical value for
a given material [82]. For example, by using a cross section design that allows for an extremely small core, suspended on three, nanometre-thin struts (similar to the design studied in Section 5.2) and employing a highly nonlinear glass (SF57), the current record in terms of the highest fibre nonlinearity was obtained at around 1860 W−1km−1 [50].
This is nearly two orders of magnitude larger than the maximum achievable with con- ventional fibres and it confirms that in the high nonlinearity regime the large index contrast and the possibility to employ a single, highly nonlinear glass are undoubtedly attractive advantages for MOFs.
The derivation of Equation 1.6 and 1.7 assumes that the nonlinear refractive index n2
does not change significantly in the transverse plane. This is typically a good assumption for all-solid conventional fibres. However, in the case of microstructured air-glass fibres,
n2 presents order of magnitude variations between the air and the glass regions (n2 of
air is∼3×10−23 m2W−1, three orders of magnitude smaller than silica). Generally it
is energetically most favourable for the mode to localise itself in the high index regions and thus only a minority of the field resides in air. For λ ¿ Λ, essentially all power is localised in silica and Equation 1.6 and 1.7 provide a reliable estimate of the fibre’s nonlinear performance. However, for λ comparable to or larger than Λ or in the case of the air-guiding PBGFs discussed in Section 1.3, a large percentage of the optical
power can be localised in air. This requires the air’s lower n2 to be appropriately be taken into account. Alternative definitions of Aeff have been proposed in order to solve this issue [82, 83]; however, in order to correctly estimate the nonlinear parameter
γ, these definitions produce a much larger and unphysical value of Aeff, a parameter usually associated with the spatial extension of the mode. In order to avoid this and to maintain the physical meaning of the effective area, in this thesis the approach suggested by Heinbergeret al. will be preferred. This redefines n2 as [84]:
n2= Z Z ∞ −∞ n2(x, y)|F(x, y)|4dxdy Z Z ∞ −∞ |F(x, y)|4dxdy (1.8)
and maintains the nonlinear coefficient definition asγ = 2πn2/(λAeff) and the definition
of Aeff as the one reported in Equation 1.7.
At the other end of the nonlinearity range, i.e. for small nonlinearity, large mode area fibres, HFs may still present some advantage over conventional fibres, but this is far less evident. Many studies have indicated that, while MOFs can have dramatically dif- ferent performances across broad wavelength ranges [85], fair comparisons at specific wavelengths show modest differences with conventional fibres in the trade-off between effective area and bending losses [86]. MOFs are generally advantaged in applications requiring single modedness over a large operational bandwidth, but for operations at a single wavelength, or where a moderately multimode output is tolerated, the debate about which is the best technology for high power delivery and high power lasers appli- cations is still ongoing.
Large mode areas would typically require a small effective index step. This can be achieved in a HF by employing claddings with very small holes and large hole separation, allowing a small decrease in the average refractive index. The largest Aeff achievable for
a single mode HF with a standard design (see Figure 1.7) and a reasonable bend loss is around 600µm2 at 1550 nm, corresponding to Λ = 23µm andd/Λ = 0.5 [87]. This can
be increased to 1400µm2 by employing just one ring of large holes (d/Λ = 0.7) around a very large core [88]. This fibre operates in an effectively single mode regime due to higher confinement and bending losses associated with higher-order modes, which can ‘leak’ through the large gaps between the holes. A more detailed analysis of this fibre however demonstrated that its effective area under bend radii tighter than 10 cm would be reduced by more than 50% and designs employing a graded index core, less susceptible to this problems were advocated [89]. Even larger modal dimensions have been achieved with a conventional HF design with a 7-rod core and an additional outer cladding of large holes. This presented an Aeff of 2000µm2 and was successfully employed to realise
an HF-based ytterbium-doped laser [57]. The bend loss issue is eliminated, in this fibre, by employing a rod-like design, resulting in a quasi-solid state configuration, with the obvious disadvantage of reduced packaging capabilities.